Calculus II · Unit 4B · lesson
Differentiating and Integrating Power Series
Differentiate and integrate power series term by term inside the radius while tracking coefficient shifts and endpoint caveats.
Section overview
Algebra and calculus with power seriesWhat this section is building
Differentiate and integrate power series term by term inside the radius while tracking coefficient shifts and endpoint caveats.
Inside a power series' convergence interval it behaves like a polynomial limit, so familiar operations become structured index shifts.
Write the source identity and validity interval before transforming coefficients or powers.
Losing an index, constant of integration, or endpoint condition during a formal manipulation.
Learning objectives
differentiate and integrate a power series term by term inside its radius of convergence.
Differentiating and Integrating Power Series
Power series behave like polynomials inside their radius
Within the open interval of convergence, a power series may be differentiated or integrated term by term. The resulting series has the same radius of convergence, although endpoint behavior can change. This theorem turns one known series into a library of new identities.
Termwise operations are powerful because differentiation and integration act simply on monomials. They are also the first place where an infinite limiting process is exchanged with another operation. The radius condition supplies the uniform control needed for that exchange on smaller closed intervals, a fact explored more carefully in analysis.
Inside the radius, a power series behaves like a polynomial
Within its radius of convergence, a power series may be differentiated or integrated term by term. Differentiation multiplies coefficients by the exponent and shifts powers downward; integration divides by the new exponent and shifts upward.
The radius remains the same, but endpoint behavior may change. Termwise operations are justified because convergence is uniformly controlled on every smaller closed interval strictly inside the radius. This prevents infinitely many small termwise errors from behaving badly when combined.
Read this graph as text
Differentiation and integration shift coefficients and powers. A row of terms maps downward under differentiation and upward under integration, with exponent multipliers or divisors displayed. Coefficient-and-exponent shifts under differentiation and integration. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in differentiation and integration shift coefficients and powers; color is never the only cue.
Why it matters: Coefficient-and-exponent shifts under differentiation and integration.
A row of terms maps downward under differentiation and upward under integration, with exponent multipliers or divisors displayed.
Differentiation and integration shift coefficients and powers. Coefficient-and-exponent shifts under differentiation and integration.
Uniform control keeps termwise operations honest
On , the power-series terms are dominated by a convergent numerical series. This uniform bound allows limits, sums, derivatives, and integrals to interact safely. The full theorem belongs to analysis, but the key is control shared by every point in the smaller interval.
Termwise calculus
If has radius , then for ,
and integration is also termwise.
Derive a logarithm series
Start from
Integrate from to :
Derive the arctangent series from geometry's favorite series
From
integrate from to :
The radius remains . Endpoint behavior must then be checked separately; at , the alternating series converges to .
Why this needs a theorem
Pointwise convergence alone does not justify interchanging a limit with differentiation. Power series possess stronger local control: on every closed interval strictly inside the radius, they converge uniformly. Uniform convergence is the analysis concept that keeps infinitely many small errors under control at once.
The radius stays, but endpoints may change
Do not copy the original interval of convergence mechanically. After differentiating or integrating, retest any finite endpoints.
u4b-differentiating_and_integrating_power_series-01Differentiate term by term.
Your work stays on this device. No account or AI grader is used.
Show hint
The constant term vanishes and .
Attempt once to unlock the solution
Submit an answer first. The hint is available now.
Integrate from to .
Differentiate the series for .
Explain why radius stays the same while endpoints may change.
Use termwise differentiation to find a series for .
Source & rights
Original instruction with traceable references.
BetterGrades-original; no direct adaptation declared in the verified handoff.
Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.